Simulation of discrete time data
18 December, 2024
Simulation 0
No association between longitudinal marker and mortality. Longitudinal marker has linear trend and independent error over time, no autoregressive component. Baseline mortality risk is time- and covariate-independent, i.e. constant over time and patients. Mortality is independent of the longitudinal trajectory.
Baseline mortality risk
Longitudinal trajectory trend
Patients dying?
## .
## FALSE TRUE
## 75604 24396## .
## 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
## 982 976 924 971 968 965 989 909 934 902 916 879 872 901 890 809
## 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14
## 897 830 813 817 790 798 818 790 802 749 719 786Patients at risk
Expected longitudinal marker trajectory for mortal and immortal cohort
Heterogeneity in longitudinal marker trajectory
Simulation 1
No association between longitudinal marker and mortality. Longitudinal marker has linear trend and independent error over time, no autoregressive component. Baseline mortality risk is time-independent but does depend on covariates. Mortality is independent of the longitudinal trajectory.
Baseline mortality risk
Longitudinal trajectory trend
Patients dying?
## .
## FALSE TRUE
## 81148 18852## .
## 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
## 735 713 746 678 724 746 741 653 737 717 735 657 619 661 683 705
## 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14
## 641 658 622 666 613 645 663 611 612 646 653 572Patients at risk
Expected longitudinal marker trajectory for mortal and immortal cohort
Heterogeneity in longitudinal marker trajectory
Simulation 2
The longitudinal marker trajectory is linear and has no autoregressive component. It is covariate dependent. The marker and mortality are positively associated based on a global dependence parameter, no local dependence changes over time. Mortality does not depend on covariates, i.e. the marker is the only component introducing association.
Baseline mortality risk
Longitudinal trajectory trend
Patients dying?
## .
## FALSE TRUE
## 53 99947## .
## 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5
## 23262 17559 13768 10526 8126 6196 4750 3591 2868 2143 1648 1289 937
## 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13
## 742 612 452 373 247 206 159 137 81 70 57 54 42
## 13.5 14
## 26 26Patients at risk
Expected longitudinal marker trajectory for mortal and immortal cohort
Heterogeneity in longitudinal marker trajectory
Simulation 3.1
The mortality risk and longitudinal marker trajectory have a quadratic trend. They are positively associated globally, without local changes.
The mortality risk is not covariate dependent.
Baseline mortality risk
Longitudinal trajectory trend
Patients dying?
## .
## FALSE TRUE
## 120 99880## .
## 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5
## 16723 14002 11848 9875 8413 6965 5885 4850 4052 3326 2741 2224 1796
## 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13
## 1433 1211 981 803 623 490 403 302 256 191 151 125 88
## 13.5 14
## 72 51Patients at risk
Expected longitudinal marker trajectory for mortal and immortal cohort
Heterogeneity in longitudinal marker trajectory
Simulation 3.2
The mortality risk and longitudinal marker trajectory have a quadratic trend. They are negatively associated globally, without local changes.
The mortality risk is not covariate dependent.
Baseline mortality risk
Longitudinal trajectory trend
Patients dying?
## .
## FALSE TRUE
## 5 99995## .
## 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5
## 31204 21623 14887 10246 7038 4865 3263 2233 1562 1029 700 442 311
## 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 13 13.5 14
## 225 136 95 55 25 27 16 5 3 2 1 1 1Patients at risk
Expected longitudinal marker trajectory for mortal and immortal cohort
Heterogeneity in longitudinal marker trajectory
Simulation 4
The longitudinal marker and the mortality probability are positively associated and depend on the covariates. Both also have a quadratic trend in time.
Baseline mortality risk
Longitudinal trajectory trend
Patients dying?
## .
## FALSE TRUE
## 157 99843## .
## 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5
## 17638 14857 11882 9256 8025 6606 5503 4679 4100 3397 2562 2279 1808
## 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13
## 1577 1310 914 776 558 513 380 323 233 202 141 107 91
## 13.5 14
## 78 48Patients at risk
### Variance across time for longitudinal marker The conditional
variance within each partition, conditional on the lag-1 longitudinal
marker value and the baseline covariates, remain constant. Considering
the marginal variance across partitions, we expect a positive increase
in variance as time progresses.
Expected longitudinal marker trajectory for mortal and immortal cohort
Heterogeneity in longitudinal marker trajectory
Simulation 5.1
The longitudinal marker has no trend but a lag-1 autoregressive component. The baseline mortality is constant.
Baseline mortality risk
Longitudinal trajectory trend
Patients dying?
## .
## FALSE TRUE
## 48523 51477## .
## 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
## 1177 1088 1159 1097 1170 1008 1236 1051 1177 1110 1101 979 1089 1028 1069 998
## 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16
## 1045 951 874 1074 858 917 860 929 916 889 830 810 814 758 831 757
## 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 21.5 22 22.5 23 23.5 24
## 750 747 788 783 825 865 800 758 728 694 779 767 812 698 642 670
## 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29 29.5 30
## 698 703 609 699 698 615 571 574 671 637 563 683Patients at risk
Variance across time for longitudinal marker
The longitudinal trajectory is stationary, as visible from the constant covariance.
Expected longitudinal marker trajectory for mortal and immortal cohort
Heterogeneity in longitudinal marker trajectory
Simulation 5.2
The longitudinal marker has no trend but a lag-1 autoregressive component. The baseline mortality is constant.
Baseline mortality risk
Longitudinal trajectory trend
Patients dying?
## .
## FALSE TRUE
## 49372 50628## .
## 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
## 1068 1156 1263 1176 1080 1099 1175 1049 1093 1033 966 990 1029 955 928 915
## 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5 16
## 1025 934 992 890 841 899 1022 822 1012 916 948 801 828 808 827 838
## 16.5 17 17.5 18 18.5 19 19.5 20 20.5 21 21.5 22 22.5 23 23.5 24
## 782 878 698 807 765 774 844 718 670 701 704 689 764 753 692 630
## 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29 29.5 30
## 660 624 599 654 669 580 606 656 612 648 552 521Patients at risk
Variance across time for longitudinal marker
The longitudinal trajectory is not-stationary, as visible from the non-constant covariance over time.
Expected longitudinal marker trajectory for mortal and immortal cohort
Heterogeneity in longitudinal marker trajectory
## [1] 500